High among today’s concerns are the potential
environmental and societal impacts of global warming. The study of global
warming is rooted in two major areas of research: (1) analysis
of massive amounts of historical climate data and (2) development of complex,
multidimensional mathematical models requiring some of the world’s largest,
fastest supercomputers.

Using some simple equations, concepts, and
assumptions, we can model the earth’s temperature, as well as the temperatures
of other planets.

Scientific
Basis

Note:
Definitions for words that appear in italic type
can be found in an online
glossary.

At earth’s mean
distance from the sun—149.6 million km—the solar energy flux
is 1.367 kW/m2. This measured quantity is called
the solar constant. By making some assumptions about how earth absorbs
this solar power, one can calculate the planet’s temperature. In the simplest
model, the planet is considered to be a blackbody. A blackbody is one
that is a perfect absorber, absorbing all radiation that falls on it. A blackbody
is also a perfect radiator. The amount of solar power that would fall on a
spherical body, e.g., earth, would be equal to the cross-sectional area of
the planet times the solar constant, S0

(1)

All matter at a temperature above absolute zero radiates
energy. The higher the temperature, the more energy is radiated. For a blackbody,
according to the Stefan-Boltzman Law, the amount of energy emitted is proportional
to the product of a constant—the Stefan-Boltzman constant, s, which has the value 5.67 x 10-8 Watts m-2 K-4—and
the fourth power of the body’s temperature,
T4.

Because spherical body would radiate over
its entire surface, the total radiated energy would also be proportional to
the surface area, 4pR2, resulting in the expression

(2)

If the planet is in a state of equilibrium, neither
heating nor cooling, the energy absorbed must be in equilibrium with the energy
radiated or emitted. In mathematical terms, expression (1) must equal expression
(2)

(3)

This equation can be solved for the blackbody planet’s
temperature, T0, which is called its effective temperature. Once
you have solved equation (4) for T, check your work by placing your mouse
pointer (cursor) in the chartreuse button below.

Modeling
Earth’s Temperature

To perform the calculations below, you should
download the Microsoft Excel spreadsheet, PlanetData.xls.

NOTE: If you do not have Excel, you can view
an image of the spreadsheet here,
print it, and perform the calculations with a calculator, preferrably
a graphing calculator.

The blue, underlined planet names are links to more
data on the planets at NASA’s Goddard Space Flight Center. You will
use these links later.

Your solution of equation (3) for T0 should have been

(4)

Blackbody earth

Calculate the blackbody temperature of earth by entering
the expression to the right of the equal sign in equation (6) as a formula
in the spreadsheet cell representing earth's effective blackbody temperature
(D8). This is the temperature the earth would be if it were
a blackbody—which of course it is not.

some questions designed allow
studentst to focus their thought and assess their understanding

Reflective earth

To refine our model, we need toconsider the fact that
the planets are not blackbodies, but that they reflect some of the incident
solar radiation (that’s why we can see them).

How would you expect a planet's temperature
to be influenced by the fact that some of the incident solar radiation is
reflected back into space? In the box below, point to the line below that correctly completes the
sentence.

The fraction of incident solar radiation that is reflected
by a planet is termed its albedo (A). For modeling a planet’s temperature,
it is the amount of energy absorbed that is of interest. The fraction of solar
radiation absorbed is (1-A), and we must modify expression (1)

(5)

and the equation for temperature becomes

(6)

Enter the expression to the right of the equals sign
as a formula in the spreadsheet (column ‘EARTH’, row ‘Effective temperature,
with albedo, K’), using the appropriate Bond albedo.

some questions designed allow
studentst to focus their thought and test their understanding

Temperatures
of Other Planets

Scientists develop numerical models
to explain how our planet behaves, and to predict how it will behave under
different conditions. The validity of these models can be tested by seeing
if they will explain how earth behaved in the past, or whether the the models
can predict how how the planet behaves in the very near future. But as varied
as earth is, it does not match the vast array of conditions on the other planets
of our solar system. So scientists turn their attention outward to learn not
just how our climate works, but how climates work, not just
how earth’s atmosphere works, but how atmospheres work.

The solar constant of other planets

Astronomers and planetary scientists refer a planet’s
distance from the sun in terms of the astronomical unit (AU), which
is the ratio between the planets mean distance from the sun and the earth’s
mean distance.

In the spreadsheet low labeled Mean distance from
the sun, AU, enter formulas to calculate each planet’s distance in terms
of the earth’s distance, i.e., earth’s distance = 1.00.

Calculate solar constant for each planet based on earth’s
solar constant and the planet’s distance from the sun using inverse square
law